Mixture Model 25

3. COMPUTATIONAL FLUID DYNAMICS MODEL EQUATIONS

3.2 TWO PHASE MODELING EQUATIONS 22

3.2.2 Mixture Model 25

The mixture model is a simplified multiphase model that can be used in different ways.

The mixture model allows us to select granular phases and calculates all properties of the granular phases. This is applicable for liquid-solid flows.

3.2.2.1 Continuity Equation

( )

+∇.

( )

=0

∂

∂

m m

t ρm ρ υ^r

(3.15) Where υ^r_mis the mass-averaged velocity:

m n

k

k k k

m ρ

υ ρ α

υ

∑

= =¹

r

r (3.16)

and ρ_mis the mixture density:

∑

=

= ⁿ

k k k m

1

ρ α

ρ (3.17)

αkis the volume fraction of phase k 3.2.2.2 Momentum Equation

The momentum equation for the mixture can be obtained by summing the individual momentum equations for all phases. It can be expressed as:

( ) ( ) [ ( ) ]



 



∇ + + +

∇ +

∇

∇ +

−∇

=

∇

∂ +

∂

∑

=

k dr k dr k n

k k m

T m m m m

m m m

m

F g t p

, , 1

.

. .

υ υ ρ α ρ

υ υ µ υ

υ ρ υ

ρ

r r r

r r r

r r

(3.18)

Where nis the number of phases, Fis a body force, and µ_mis the viscosity of the mixture and defined as

∑

= _{k k}

m α µ

µ (3.19)

k

υ^rdr, is the drift velocity for secondary phase k:

m k k

dr υ υ

υ^r _, = ^r − ^r (3.20)

3.2.2.3 Energy Equation

The energy equation for the mixture takes the following form:

( ) ( ( ) ) (

eff

)

E

n

k

k k k k n

k

k k

k E E p k T S

t +∇ + =∇ ∇ +

∂

∂

∑ ∑

=

=

. .

1 1

ρ υ α ρ

α ^r (3.21)

Where k_eff is the effective conductivity

( _∑

αk⁽kk ⁺kt⁾

)

, where k_tis the turbulent thermal conductivity, defined according to the turbulence model being used). The first term on the right-hand side of Eq. 3.21 represents energy transfer due to conduction. S_E include any other volumetric heat sources. In Eq. 3.21

2

2 k k k k

p v h

E = − +

ρ ^(3.22)

for a compressible phase, and E_k =h_k for an incompressible phase, where h_kis the sensible enthalpy for phase k .

3.2.2.4 Volume Fraction Equation for the Secondary Phases

From the continuity equation for secondary phasep, the volume fraction equation for secondary phase pcan be obtained:

( ) ( ) ( ) _∑

= 















− +

−∇

=

∇

∂ +

∂ ⁿ

q

pq qp p

dr p p m

p p p

p m m

t ₁

. .

. ,

.α ρ υ α ρ υ

ρ

α ^{r r} (3.23)

Chapter 4

SIMULATION OF SINGLE PHASE FLUID FLOW IN A CIRCULAR MICRO CHANNEL

It is well known that nanoparticles have very high thermal conductivity compared to commonly used coolant. Thus, the thermal conductivity and other fluid properties are changed by mixing the particle in fluid. The changed properties of the nanofluids determine the heat transfer performance of the microchannel heat exchanger with nanofluids. This point is illustrated in this chapter by doing the computational fluid dynamics (CFD) analysis of the hydrodynamics and thermal behaviour of the single phase flow through a circular micro channel (Lee and Mudawar, 2007).

4.1 SPECIFICATION OF PROBLEM

Consider a steady state fluid flowing through a circular micro channel of constant cross- section as shown in Fig. 4.1 (Lee and Mudawar, 2007). The diameter and length of circular micro channel are 0.0005m and 0.1m respectively. The inlet velocity is u (m/s), which is constant over the inlet cross-section. The fluid exhausts into the ambient atmosphere which is at a pressure of 1 atm.

Figure 4.1

:

Fluid flow through a circular micro channel of constant cross-section As fluid flows through in a pipe at both hydraulic and thermally fully developed condition, the Nusselt number is constant for laminar flow and it follows the Dittius- Boelter equation for turbulent flow.

oC

in in

30 T

kPa 100 P

=

=

q^// =100 W/m²

Pout = 100 kPa q^// =100 W/m²

D=0.0005 m



=

flow) (turbulent g/s

0 . 4

flow) (laminar

g/s 4 . 0

.

m

Nu = 4.36 for laminar flow (4.1)

36 . 4

i.e. =

k

hD ⇒

h ≈ κ

(4.2)

And

4 . 0 8 .

0 Pr

Re 023 .

=0

Nu for turbulent flow (4.3)

k i.e. hD =

4 . 8 0

. 0

023 .

0 











 





k Dv Cpµ

µ

ρ

4 . 0 8 . 0 6 .

0 −

≈

⇒ h κ υ µ

(4.4)

From Eq. 4.2 and Eq. 4.4 it is clear that thermal conductivity has greater effect on heat transfer coefficient for laminar flow as compared to turbulent flow. This implies the enhancement effect due to the increased thermal conductivity of nanofluids is significantly weaker for turbulent flow than for laminar. The enhancement in turbulent flow is also dependent on flow rate in addition to viscosity and specific heat. Since

4 . 0 4 . 0 6 . 0

Cp

h≈κ µ⁻ and because increased nanoparticle concentration enhances viscosity and degrades specific heat, the enhancement effect of nanoparticles in turbulent flow is further reduced compared to thermal conductivity alone.

4.2 GEOMETRY IN ANSYS WORKBENCH

The Computational domain of circular micro channel is represented in two dimensional (2D) form by a rectangle and displayed in Fig. 4.2. The geometry consists of a wall, a centerline, and an inlet and outlet boundaries. The radius, R and the length, L ofthe pipe are specified in the figure.

Figure 4.2: Computational Domain of Circular Micro channel Wall

Axis

L=0.1 m

Inlet R=0.00025 m Outlet

4.3 MESHING OF GEOMETRY

Structured meshing method done in ANSYS Workbench was used for meshing the geometry. 100×10 nodes were created. The 2D geometry of circular micro channel with structured mesh is shown in Fig. 3.

Figure 4.3: Two dimensional geometry of circular micro channel with structured mesh

4.4 PHYSICAL MODELS

Based on the Reynolds number,Re=Duρ µ, either viscous laminar model or standard ε

κ − model is used for laminar and turbulent flow respectively. The choice of the model is shown in Table 4.1. D is the diameter of the microchannel, ρ and µare the density and viscosity of the fluid.

Table 4.1: Choice of model based on Reynolds number

Reynolds no. (Re) Flow (Model)

< 2000 Laminar

> 2000 κ −ε Model

4.5 MATERIAL PROPERTIES

Pure water is used as base working fluid and Alumina (Al2O3) is taken as nanoparticles.

The density, heat capacity and thermal conductivity of alumina are 3,600 kg/m³, 765 J/kgK and 36 W/mK respectively. The properties of nanofluids (nf) are given in Table 4.2 at 30^oC temperature and 100 kPa pressure.

Table 4.2 Water base fluid properties with different concentration of alumina nanoparticles (Lee and Mudawar, 2007)

φ=0% φ=1% φ=2% φ=3% φ=4% φ=5%

knf(W/mK) 0.603 0.620 0.638 0.656 0.675 0.693

ρnf(kg/m³) 995.7 1021.7 1047.7 1073.8 1099.8 1125.9 µnf(kg/m s) 7.97X10^-4 8.17X10^-4 8.376X10^-4 8.576X10-⁴ 8.775X10^-4 8.974X10-⁴

nf

Cp_, (kJ/kg K) 4.183 4.149 4.115 4.081 4.046 4.012

4.6 GOVERNING EQUATIONS

For 2D axis symmetric geometries, the continuity equation is given by (ANSYS Fluent 12.0).

( ) ( )

+ =0

∂ + ∂

∂ + ∂

∂

∂

r r

x t

r r

x

ρυ ρυ ρ ρυ

(4.5) Where x represents axial coordinate in the direction of flow, r is the radial coordinate i.e. transverse direction, υ_x is the axial velocity, and υ_r is the radial velocity components.

For 2D axis symmetric geometries, the axial and radial momentum conservation equations are given by (ANSYS Fluent 12.0)

( ) ( )

( )

_x

x r x

x

x r x

x x

dv F r r

r r r x

x r

x r p

r r r

x r t

 +



 











 ∂

∂ +

∂

∂ + ∂



 



 



 



 − ∇

∂

∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

∂

υ µ υ

υ υ µ

υ ρυ υ

ρυ ρυ

. 1 3 2 2

1

1 ) 1

(

r

(4.6)

And

( ) ( ) ( )

( )

^r

( )

^z _r

r

x r r

r r

x r

r F r r

r r r r

r r x

x r r r p

r r r

x r t

+ +

∇ +

−



 



 



 



 − ∇

∂

∂

∂ + ∂



 



 



 





∂ +∂

∂

∂

∂ + ∂

∂

−∂

∂ = + ∂

∂ + ∂

∂

∂

2

2 .

3 2 2

3 . 2 2

1

1 1

1

ρυ µ υ

µυ υ υ

µ

υ µ υ

υ υ υ

υ ρυ

r

r (4.7)

Where,

r r x

r r

x υ υ

υ υ +

∂ +∂

∂

=∂

∇ r .

(4.8) Since the microchannel with small radial thickness is horizontally placed, the external body force F is taken as zero.

Turbulent flows are characterized by fluctuating velocity fields. These fluctuations mix transported quantities such as momentum, energy, and species concentration, and cause the transported quantities to fluctuate as well. Since these fluctuations can be of small scale and high frequency, they are too computationally expensive to simulate directly in practical engineering calculations. Instead, the instantaneous (exact) governing equations can be time-averaged, ensemble-averaged, resulting in a modified set of equations that are computationally less expensive to solve. However, the modified equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities.

The standard k −ε model (ANSYS Fluent 12.0) is used to model single phase turbulent flow in circular micro channel. The turbulence kinetic energy,k, and its rate of dissipation, ε , are obtained solving the following transport equations:

( ) ( )

_{k b M k}

j k

t j

i i

S Y G

x G k k x

k x

t + + − − +













∂

 ∂







 +

∂

= ∂

∂ + ∂

∂

∂ ρε

σ µ µ υ

ρ

ρ (4.9)

And

( )

_ε

(

_ε

)

_{ε ε}

ε

ρ ε ε

ε σ µ µ ρευ

ρε S

C k G C k G

x C x

x

t _j ^{k b}

t j

i i

+

− +

+











∂

 ∂



 



 +

∂

= ∂

∂ + ∂

∂

∂ ²

2 3

) 1

( (4.10)

In these equations, G_krepresents the generation of turbulence kinetic energy due to the mean velocity gradients. G_b is the generation of turbulence kinetic energy due to buoyancy. Y_M represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate.C_1ε, C_2εand C_3εare constants. σ_k and σ_εare the turbulent Prandtl numbers for kandε , respectively. S_kand S_εare user-defined source terms.

Modelling the Turbulent Viscosity

The turbulent (or eddy) viscosityµ_t is computed by combining kand ε as follows:

ρ ε

µ_t = C_µ ^k2 (4.11)

Model Constants

The model constant C_1ε.C_2ε, C_3ε σ_kand σ_εhave the following default valuesC_1ε =1.44,C_2ε =1.92, C_µ =0.09,σ_k =1.0and σ_ε =1.3

The governing equation for energy is same as represented in Eq. 3.4.

The bulk mean temperature,

T

_m,x and wall temperature,

T

_w,x with distance x from the microchannel entrance can be obtained by doing the thermal energy balance around the microchannel as shown in Fig. 1. The equations are Eq. 4.12 and Eq. 4.13. .

nf p in

x m

C m

Dx T q

T

, .

"

,

+ π

=

_(4.12)

h T q

T

_{wx mx}

"

,

,

= +

_(4.13)

Where,T_in (303.15 K), is the specified inlet temperature. q^"and h are the heat flux and heat transfer coefficient respectively.

To characterize the effect of fluid flow on the thermal behaviour of the microchannel heat exchanger Peclet number, Pe is defined as

L k

u Pe ρ^fC^p,f

= Conductive Flux Flux Convective

= (4.14)

4.7 BOUNDARY CONDITIONS

A no slip boundary condition was assigned for the non porous wall surfaces, where both velocity components were set to zero at that boundary i.e.

υ

_x

= υ

_r

= 0

. A constant heat flux (100 W/m2) is applied on the channel wall. Axis symmetry was assigned at centerline. A uniform mass flow inlet and a constant inlet temperature were assigned at the channel inlet. At the exit, pressure was specified.

4.8 METHOD OF SOLUTIONS

There are two ways to solve the problem stated above: (i) analytical method and (ii) CFD method. Lee and Mudwar, 2007 have used analytical method. The method consists of calculation of heat transfer coefficient (h) from either of Eq. 4.3 or Eq. 4.4 depending on whether flow is laminar or turbulent, calculation of bulk mean temperature of fluid using Eq. 4.12 followed by the calculation of wall temperature using eq. 4.13. The results given by Lee and Mudwar, 2007 are having errors in wall temperature and mean temperature calculation at the given inlet mass flow rate. Hence, in the present study, the analytical values of heat transfer coefficients are calculated using Eq. 4.3 / Eq. 4.4. The heat transfer coefficients are also obtained using CFD methods and then both the values are compared.

The CFD method follows the use of commercial software ANSYS Fluent 12.0 to solve the problem. The specified solver in Fluent uses a pressure correction based iterative SIMPLE algorithm with 1st order upwind scheme for discretising the convective transport terms. The convergence criteria for all the dependent variables are specified as 0.001. The default values of under-relaxation factor as shown in Table 4.3 are used in the simulation work.

Table 4.3 Relaxation factor

Factors Value

Pressure 0.3

Density 1.0

Body force 1.0

Momentum 0.7

Energy 1.0

Using the CFD computed wall temperatures and mean temperature from Eq. 4.12; heat transfer coefficients can be calculated by Eq. 4.13.

4.9 RESULTS AND DISCUSSIONS

The hydrodynamics behaviour of the channel can be studied in terms velocity distribution within the channel. The value of Re is 1278.0 for inlet mass flow rate 0.4 gm/s and it becomes 12780.0 for 4.0 gm/s inlet mass flow rate. Thus the flow is laminar and turbulent for 0.4 gm/s and 4.0 gm/s inlet mass flow rates respectively. The variation of centreline velocity at laminar state with axial position (X) for water and its nanofluid are displayed in Fig. 4.4. The figure shows that for all the fluids the entrance lengths i.e. the length required to reach fully developed state are the same. The density and viscosity of nanofluids increases with increase in the nanoparticles concentration in water. Thus, the velocity at any axial position decreases with increase in the nanoparticles concentration as found in Fig. 4. The same is shown at turbulent state in Fig. 4.5. It is also observed here that the axial velocity decreases with increase in nanofluid concentrations. Unlike laminar flow, the entrance lengths are found different in turbulent flow condition.

Figure 4.4: Velocity profile at centerline in the circular micro channel at Re =1278. PrWater

=5.53 , Pr2% = 5.40, Pr5% = 5.20.

Figure 4.5: Velocity profile at centerline in the circular micro channel at Re =12780.

PrWater =5.53 , Pr2% =5.40 , Pr5% =5.20.

Comparison of analytical and computational heat transfer coefficient for laminar (Re = 1278.0) and turbulent flow (Re = 12780.0) of water and its nanofluids (with 2% and 5%

alumina) are shown in Table 4.4 and 4.5 respectively. The same comparison is also displayed in Fig. 4.6 and 4.6. The comparison shows that the CFD results can well predict the analytical heat transfer coefficient. The heat transfer data shows that inclusion of nanoparticles in water increases the heat transfer coefficient. More nanoparticle concentration more is the heat transfer coefficient. In case of laminar flow, both the analytical and CFD results show that heat transfer coefficient increases approximately by 15% magnitude from pure water to 5% nanofluid. As expected, both the tabular data and figures show that heat transfer coefficients in turbulent flow are more than laminar values.

Like laminar heat transfer coefficient, the values of heat transfer coefficients in turbulent flow also increases with increase in nanoparticles concentration. Both the analytical and CFD values of it increase by 12% magnitude from pure water to 5% nanofluid. Thus, the increase of heat transfer coefficient is more in laminar flow than turbulent flow. But the percentage increase in heat transfer coefficient in turbulent zone is not negligible. These results contradict the results obtained by Lee and Mudawar, (2007). The use of nanofluids favourable therefore favourable both in laminar and turbulent fluid flow regimes. The heat transfer coefficient values for both type of flow are found to be independent of axial position. It means that circular micro channel is at fully thermal developed condition in both cases.

Table 4.4Comparison of analytical heat transfer coefficient with the present CFD results at laminar (Re == 1278.0) flow for water and nanofluid.

Analytical Heat Transfer Coefficient(W/m².K)

Computational Heat Transfer Coefficient(W/m².K) Position

(m)

Pure Water

Nanofluid with 2 % Alumina

Nanofluid with 5 % Alumina

Pure Water

Nanofluid with 2 % Alumina

Nanofluid with 5 % Alumina

0 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

0.02 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

0.04 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

0.06 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

0.08 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

0.1 5258.16 5563.36 6042.96 5263.16 5555.56 6044.21

Table 4.5Comparison of analytical heat transfer coefficient with the present CFD results at turbulent (Re =12780.0) flow for water and nanofluid.

Analytical Heat Transfer Coefficient(W/m².K)

Computational Heat Transfer Coefficient(W/m².K) Position

(m)

Pure Water

Nanofluid with 2 % Alumina

Nanofluid with 5 % Alumina

Pure Water

Nanofluid with 2 % Alumina

Nanofluid with 5 % Alumina 0 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5 0.02 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5 0.04 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5 0.06 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5 0.08 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5 0.1 106020.5 111111.9 118882.3 106097.2 111184.5 118893.5

Figure 4.6: Variation heat transfer coefficient for laminar flow (Re =1278.0) in circular micro channel for water and its nanofluid

Figure 4.7: Variation heat transfer coefficient for turbulent flow (Re =12780.0) in circular micro channel for water and its nanofluid

In laminar flow Nu=4.36(i.e. h∝k). To prove it, Table 4.6 shows the ratio of heat transfer coefficient of water to heat transfer coefficient of nanofluids at different values of Re. From Table it is clear that the ratio of heat transfer coefficient of water to heat transfer coefficient of nanofluids is equal to thermal conductivity of water to thermal conductivity its nanofluid. Hence, at laminar flow conditionh∝k.

Table 4.6: Validation of laminar flow model based on the relation between heat transfer coefficient and thermal conductivity of the fluid.

The variation of wall temperature with axial distant (X) is shown in Fig. 4.8, 4.9 and 4.10.

The analytical wall temperature can be calculated from Eq. 4.13 using the anlytical values

of heat transfer coefficiet. In Fig. 4.8 and 4.9 analytical wall temperatures are also compared with the wall temperatures obtained using CFD methods. It is oberved that both the ananltycal and compted wall temperatures are same at all position. The figures also depicts that the wall temperatures are equal to inlet temperatures over the axial distane of the channel. It happents both in the laminar and turbulent flow conditions. The value of Peclet number, Pe for both Re = 1278.0 and Re = 12780.0 are much much greater than 1.0, and hence temperature at all the point on axis becomes equal to inlet temperature. As Re decreass to 0.1 (very low value), the Pe value decreases substantially and also residenc time of the fluid in the channel increases due to decrease in inlet velocity of the fluid.

Thus a substantial change in temperature from inlet to outlet of the channel is observed at

Re Heat

Transfer Coefficient

(W/m².K)

06 . 1

% 0

%

2

=

k

k 1 . 15

% 0

%

5

=

k

k 1 . 09

% 2

% 5

= k k

%

h0 h_2% h_5%

% 0

% 2

h h

% 0

% 5

h h

% 2

% 5

h h

0.1 558.65 591.06 642.1 1.058 1.149 1.09

5 3333.3 3533.3 3826.5 1.06 1.148 1.08

25 4545.4 4805.6 5227.1 1.058 1.151 1.09

50 5003 5284.2 5758.5 1.056 1.151 1.09

Re = 0.1 shown in Fig. 4.10. It is also noticed in Fig. 4.10 that the use of pure water results in more wall temperature than its nanofluids, and wall temperature at a particular axial position decreases with increase in nano particle concentrations. It might happen due to combined effect of density, viscosity and thermal conductivity.

Figure 4.8: Variation wall temperature for laminar flow (Re =1278.0) in circular micro channel for water and its nanofluid

Figure 4.9: Variation wall temperature for turbulent flow (Re =12780.0) in circular micro channel for water and its nanofluid

Figure 4.10: Variation wall temperature for Re = 0.1 in circular micro channel for water and its nanofluid

The study on variation of wall temperature using water as the heating medium in circular micro channel is done for different values of Reynolds number. These are tabulated in Table 4.7 and also depicted in Fig. 8. From the figure it is clear that there is less variation in wall temperature at higher Re than the lower one. The table and figure also depicts that effectively there is no change in temperature with distance as Re is greater than 1.0. The Peclet number,Pe increases with increase in Re. Thus the contribution of convective heat flux dominates over conductive heat flux at higher Re. The inlet temperature of the fluid penetrates more towards the outlet at higherPe. Hence, it is expected that the variation of temperature reduces from inlet to outlet with increase in Re.

The effect of Reynolds number on the variation of wall temperature with axial position is represented in Table 4.8 and also in Fig. 4.11. It shows that as Re decreases the wall temperature increases more with axial position. It occurs due to decrease in Pe with the decrease in Re.

Table 4.7 Variation of wall temperature with axial distance at different values of Re

Wall Temperature K

Position (m)

Re=0.1 Re=1 Re=5 Re=10 Re=15 Re=25 Re=30 Re=50

0 303.32 303.12 303.035 303.022 303.018 303.019 303.012 303.004

0.02 327.239 305.47 303.511 303.265 303.183 303.117 303.101 303.068

0.04 351.241 307.87 303.991 303.505 303.343 303.213 303.181 303.116

0.06 375.24 310.27 304.471 303.745 303.503 303.309 303.261 303.164

0.08 399.32 312.67 304.951 303.985 303.663 303.405 303.341 303.212

0.1 421.42 315 305.419 304.219 303.819 303.499 303.419 303.258

Figure 4.11: Variation of wall temperature with axial distant for different value of Reynolds number and Peclet number. Water is used as the fluid in the heat exchanger.

Varition of velocity of water with radius (Y) at different axial position (X) for diffrent values of Re is shown in Fig. 4.12. From the figure its is clear that velocity has no variation with X i.e. from inlet to exit the profiles are same. This indicates that ciruclar mirochannel is at fully developed flow condition.The same is true for all fluid flow with water as well as its nanofluid. One typical obsevation is made that with increasing Re, the velocity profile becomes more parabolic. The variation of velocity is found to be limited closer to the wall and around the center a flat profile is observed at Re = 0.1. This undeveloped velocity at lower Re results in domination of conductive heat transport compared to the higher Re case, and thus larger variation of temperatures are occuring at lower Re.

(a)

(b)

(c)

Figure 4.12: Variation of velocity of water in radial direction at different values of X for (a) Re = 0.1 (b) Re = 25 (c) Re = 50. Water is used as the fluid in the heat exchanger.

The varition of wall temperature of water with radial posititon (Y) at different values of X and at different values of Re is shown in Fig. 4.13. All the figure shows that there is no variation of temperature in Y direction. It happens because convective heat transfer rate dominates over conductive heat transfer rate even at Re = 0.1 for which also Pe >> 1. At Re = 0.1, temperature increases with the increase X, and the possible explanation is given while discussing the velocity profile of water in the exchanger. The relative comaprison of variation of temperature at different Re depicts that outlet temperature approaches inlet temperature with increase in Re. It occurs due to increase in Peclet number with Re.

Fig. 4.13(c) represents the variation wall temperature with Y at different X at very low value of Re. In this case, Pe is much less than 1.0. Thus conductive heat transport dominitates over convective heat transfer here. But still there is no temperature distribution observed with radial position. It happens because the the diameter of the channel is very very less compared to the length.

(a)

(b)

(c)

Figure 4.13: Temperature of pure water at different position in X direction for diffrent values of Re (a) Re = 0.1 (b) Re = 25 (c) Re = 50. Water is used as the fluid in the heat exchanger.

To see the entrance effect of circular micro channel on the velocity profile a velocity contours plot for differnt Reynolds are shown in Figs. 4.14 to 4.16. The figures show that the fluid rquires to travel certain distance in the flow direction called entrance length to get fully developed velocity profile. It is quite viscible in the figures that the entrance length as expected increases with increase in Reynolds number. The velocity contours at the exit also show that wall effect penetrates more towards the center and the thicknes of the zone with maximum velocity shrinks with increase in Re.

(a)

(b)

Figure 4.14: Water velocity contour plot at Re = 0.1 for water (a) Nearer to the entrance (b) Nearer to the exit